## Section: New Results

### Absorbing boundary conditions and absorbing layers

#### Perfectly matched transmission problem with absorbing layers : application to anisotropic acoustics

Participant : Sébastien Impériale.

This work has been carried out in collaboration with Edouard Demaldent from CEA-LIST. We have worked on an original approach to design perfectly matched layers (PML) for transient wave equations. This approach is based, first, on the introduction of a modified wave equation and, second, on the formulation of general “perfectly matched” transmission conditions for this equation. The stability of the transmission problem is discussed by way of the adaptation of a high frequency stability (necessary) condition, and we apply our approach to define PML suited for the anisotropic wave equation. A variational formulation of the problem is then developed. It includes a Lagrange multiplier at the interface between the physical and the absorbing domains to deal with the “perfectly matched” transmission conditions. We have carried out numerical results in 2D and 3D that first show the validity of our approach in term of stability and accuracy and the efficiency when using constant damping coefficients combined with high order elements. This work has been submitted for publication.

#### On the stability in PML corner domains

Participant : Eliane Bécache.

In collaboration with Andres Prieto, from the university of Santiago de Compostella.

We have finalized our work on the stability of the discretization of PMLs in the corners and submitted a paper (see preprint [24] ).

#### High-order Absorbing Boundary Conditions for anisotropic elastodynamics

Participants : Daniel Baffet, Eliane Bécache.

This work is done in collaboration with Daniel Baffet, PhD student of Dan Givoli, at the Technion University in Haifa (Israel) and has started during a visit of Daniel at Poems.

The aim is to design new efficient and stable absorbing boundary conditions for anistropic materials. It is known that the anisotropy introduces a specific difficulty, for ABC as well as for PMLs, in particular for models which involve inverse modes, i.e. waves for which the phase velocity and the group velocity propagate in opposite directions (with respect to the boundary). This has given rise to specific treatment for scalar models but for anisotropic elastodynamics, there are some materials for which no satisfactory solution exist. for these materials however, we can design a low order boundary condition, which is proved to be stable via an energy estimate. We have started to investigate several ways to design higher-order boundary conditions. The main difficulty is to show whether these boundary conditions are stable or not...

#### Dirichlet to Neumann map with overlap for waveguides

Participants : Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss, Geoffrey Martinache, Antoine Tonnoir.

For scattering problems in acoustic waveguides, a usual approach consists in restricting the computation to a bounded domain containing the sources and the perturbations, using transparent boundary conditions on the artificial boundaries. These conditions are written by using the so-called Dirichlet-to-Neumann maps which can be expressed thanks to a modal decomposition.

An iterative solution of the related linear system can be seen as a domain decomposition formulation without overlapping, where one domain is the bounded region and the other one is infinite. This iterative method does not converge necessarily. A classical idea is to consider a domain decomposition method with overlapping. In this work, we find the equivalent of this method in terms of a new Dirichlet-to-Neumann operator which links the trace of the solution on a section of the waveguide to the normal trace on a different one. This operator can also be expressed analytically via a modal decomposition. Its main advantage is that, because of the overlapping, it becomes compact and this is exactly why we think an iterative resolution has more chance to converge. Other advantages will appear with the elasticity application. Indeed, in the formulation of the transparent boundary condition without overlapping, appears a lagrange multiplier which makes the resolution more costly. This additional unknown will be avoided with an overlap.

For now, the theory is done for the scalar acoustic waveguide and the method has been implemented in the Melina code. The extension to the elastic case is in progress.

#### An alternative to DTN maps in waveguides

Participants : Anne-Sophie Bonnet-Ben Dhia, Guillaume Legendre.

We are interested by the treatment of the radiation condition at infinity for the numerical solution of a problem set in an unbounded waveguide. We have proposed an alternative to the classical approach involving a modal expression of Dirichlet-to-Neumann (DtN) operators. This new method is particularly simple to implement since it only requires to solve several times a boundary value problem with local boundary conditions. In the case of a an acoustic waveguide, we have proved that the corresponding approximate solution is comparable in accuracy to the one obtained by truncating the infinite series in the DtN maps. The number of linear systems to invert, which has to be greater than the number of propagative guided modes, can be significantly reduced by combining the approach with the perfectly matched layer (PML) technique. It works even in elastic waveguides, despite existence of the so-called backward waves which are known to make the PMLs inefficient, when used alone.